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Binary Goppa Code
In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups. Construction and properties A binary Goppa code is defined by a polynomial g(x) of degree t over a finite field GF(2^m) with no repeated roots, and a sequence L_1, ..., L_n of n distinct elements from GF(2^m) that are not roots of g. Codewords belong to the kernel of the syndrome function, forming a subspace of \^n: : \Gamma(g,L)=\left\ The code defined by a tuple (g,L) has dimension at least n-mt and distance at least 2t+1, thus it can encode messages of length at least n-mt using codewords of size n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternant Code
In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes. Definition An ''alternant code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of alternant form ''H''''i'',''j'' = αji''y''''i'', where the α''j'' are distinct elements of the extension GF(''q''''m''), the ''y''''i'' are further non-zero parameters again in the extension GF(''q''''m'') and the indices range as ''i'' from 0 to δ − 1, ''j'' from 1 to ''n''. Properties The parameters of this alternant code are length ''n'', dimension ≥ ''n'' − ''m''δ and minimum distance ≥ δ + 1. There exist long alternant codes which meet the Gilbert–Varshamov bound. The class of alternant codes includes * BCH code In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Code Rate
In telecommunication and information theory, the code rate (or information rateHuffman, W. Cary, and Pless, Vera, ''Fundamentals of Error-Correcting Codes'', Cambridge, 2003.) of a forward error correction code is the proportion of the data-stream that is useful (non- redundant). That is, if the code rate is k/n for every bits of useful information, the coder generates a total of bits of data, of which n-k are redundant. If is the gross bit rate or data signalling rate (inclusive of redundant error coding), the net bit rate (the useful bit rate exclusive of error correction codes) is \leq R \cdot k/n. For example: The code rate of a convolutional code will typically be , , , , , etc., corresponding to one redundant bit inserted after every single, second, third, etc., bit. The code rate of the octet oriented Reed Solomon block code denoted RS(204,188) is 188/204, meaning that redundant octets (or bytes) are added to each block of 188 octets of useful information. A few er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCH Codes
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called ''Galois field''). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D.K. Ray-Chaudhuri. The name ''Bose–Chaudhuri–Hocquenghem'' (and the acronym ''BCH'') arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri). One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niederreiter Cryptosystem
In cryptography, the Niederreiter cryptosystem is a variation of the McEliece cryptosystem developed in 1986 by Harald Niederreiter. It applies the same idea to the parity check matrix, H, of a linear code. Niederreiter is equivalent to McEliece from a security point of view. It uses a syndrome as ciphertext and the message is an error pattern. The encryption of Niederreiter is about ten times faster than the encryption of McEliece. Niederreiter can be used to construct a digital signature scheme. Scheme definition A special case of Niederreiter's original proposal was broken but the system is secure when used with a Binary Goppa code. Key generation #Alice selects a binary (''n'', ''k'')-linear Goppa code, ''G'', capable of correcting ''t'' errors. This code possesses an efficient decoding algorithm. #Alice generates a (''n'' − ''k'') × ''n'' parity check matrix, ''H'', for the code, ''G''. #Alice selects a random (''n'' − ''k'') × (''n'' − ''k'') binary invertible matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptosystem
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term ''cipher'' (sometimes ''cypher'') is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term ''cryptosystem'' is most often used when the key generation algorithm is important. For this reason, the term ''cryptosystem'' is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques. Formal definition Mathematically, a cryptosystem or encryption scheme can be defined as a tuple (\mathcal,\mathcal,\mathcal,\mathcal,\mathcal) with the following properties. # \mathcal is a set called the "plaintext space". Its elements are called plaintexts. # \mathcal is a set called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post-quantum
In cryptography, post-quantum cryptography (sometimes referred to as quantum-proof, quantum-safe or quantum-resistant) refers to cryptographic algorithms (usually public-key algorithms) that are thought to be secure against a cryptanalytic attack by a quantum computer. The problem with currently popular algorithms is that their security relies on one of three hard mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem. All of these problems could be easily solved on a sufficiently powerful quantum computer running Shor's algorithm. Even though current quantum computers lack processing power to break any real cryptographic algorithm, many cryptographers are designing new algorithms to prepare for a time when quantum computing becomes a threat. This work has gained greater attention from academics and industry through the PQCrypto conference series since 2006 and more recently by several workshops on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert–Varshamov Bound
In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert and independently Rom Varshamov.) is a limit on the parameters of a (not necessarily linear) code. It is occasionally known as the Gilbert– Shannon–Varshamov bound (or the GSV bound), but the name "Gilbert–Varshamov bound" is by far the most popular. Varshamov proved this bound by using the probabilistic method for linear codes. For more about that proof, see Gilbert–Varshamov bound for linear codes. Statement of the bound Let :A_q(n,d) denote the maximum possible size of a ''q''-ary code C with length ''n'' and minimum Hamming distance ''d'' (a ''q''-ary code is a code over the field \mathbb_q of ''q'' elements). Then: :A_q(n,d) \geqslant \frac. Proof Let C be a code of length n and minimum Hamming distance d having maximal size: :, C, =A_q(n,d). Then for all x\in\mathbb_q^n , there exists at least one codeword c_x \in C such that the Hamming distance d(x,c_x) between x and c_x satisfies :d(x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Euclidean Algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's identity, which are integers ''x'' and ''y'' such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are coprime. With that provision, ''x'' is the modular multiplicative inverse of ''a'' modulo ''b'', and ''y'' is the modular multiplicative inverse of ''b'' modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end\right/math>. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with ''n'' columns and ''n'' rows is diagonal if \forall i,j \in \, i \ne j \implies d_ = 0. However, the main diagonal entries are unrestricted. The term ''diagonal matrix'' may s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde Matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x_3 & x_3^2 & \dots & x_3^\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & x_m & x_m^2 & \dots & x_m^ \end, or :V_ = x_i^ \, for all indices and . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix is called a ''Vandermonde polynomial'' or ''Vandermonde determinant''. Its value is the polynomial :\det(V) = \prod_ (x_j - x_i) which is non-zero if and only if all x_i are distinct. The Vandermonde determinant was sometimes called the ''discriminant'', although, presently, the discriminant of a polynomial is the square of the Vandermonde determinant of the roots of the polynomial. The Vandermonde determinant is an alternating form in the x_i, meaning that exchang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
